Physics-Informed High-order Graph Dynamics Identification Learning for Predicting Complex Networks Long-term Dynamics

TL;DR

This work tackles long-horizon prediction of complex networks by modeling dynamic, high-order, non-pairwise interactions and enforcing physical consistency. It introduces PhyHSL, a framework that combines dynamic hypergraph structure learning with a dual-drive prediction scheme: physics-informed neural ODEs for continuous evolution and Koopman operator-based linearization for robust, global dynamics. The approach is trained via variational inference and demonstrated on public networks and industrial supply chains, outperforming GNN, hypergraph, and PINN baselines in accuracy and long-term stability. The results suggest substantial practical impact for reliable forecasting in real-world systems where higher-order interactions and physical constraints are essential.

Abstract

Learning complex network dynamics is fundamental to understanding, modelling and controlling real-world complex systems. There are two main problems in the task of predicting the dynamic evolution of complex networks: on the one hand, existing methods usually use simple graphs to describe the relationships in complex networks; however, this approach can only capture pairwise relationships, while there may be rich non-pairwise structured relationships in the network. First-order GNNs have difficulty in capturing dynamic non-pairwise relationships. On the other hand, theoretical prediction models lack accuracy and data-driven prediction models lack interpretability. To address the above problems, this paper proposes a higher-order network dynamics identification method for long-term dynamic prediction of complex networks. Firstly, to address the problem that traditional graph machine learning can only deal with pairwise relations, dynamic hypergraph learning is introduced to capture the higher-order non-pairwise relations among complex networks and improve the accuracy of complex network modelling. Then, a dual-driven dynamic prediction module for physical data is proposed. The Koopman operator theory is introduced to transform the nonlinear dynamical differential equations for the dynamic evolution of complex networks into linear systems for solving. Meanwhile, the physical information neural differential equation method is utilised to ensure that the dynamic evolution conforms to the physical laws. The dual-drive dynamic prediction module ensures both accuracy and interpretability of the prediction. Validated on public datasets and self-built industrial chain network datasets, the experimental results show that the method in this paper has good prediction accuracy and long-term prediction performance.

Figures (8)

• Figure 1: Examples of dynamic higher-order graph structures in supply chains and transport networks. (a) In a supply chain network, the common supply relationship of multiple parties constitutes an unpaired relationship between firms li2023learning. (b) In a transport network, both commercial and residential areas may imply static hyperedges, while external events may bring about dynamic hyperedges zhao2023dynamic.
• Figure 2: The framework of PhyHSL. (a) Firstly, we model the complex dynamic network as a temporal graph, and then capture the high-order adjacency relationship between nodes by first-order spatial convolution and second-order spectral domain convolution. Then, in order to study the high-order non-pairwise interaction between complex network nodes, dynamic hypergraph structure learning is introduced to generate each hyperedge embedding by aggregating the information of all connected nodes, and then the hypergraph embedding is used to update the node embedding to achieve high-order correlation learning of complex networks. Then we propose the physical information data dual driving strategy for network dynamic learning. In the physical information driven module, we introduce the nonlinear dynamic differential equation of information diffusion in the graph, and use the ODE solver to obtain the future state. In the data-driven module, we introduce the Koopman operator to transform the nonlinear dynamics into linear dynamics, so as to predict the future state. Finally, the two parts were fused and the final predicted network dynamics was obtained through the two-layer MLP.
• Figure 3: Dynamic hypergraph structure learning module.We first generate a low-rank correlation matrix using the node features. The hypergraph convolution operation first fuses the information from the connected nodes into the hyperedge representation and then reconstructs the node representation using the associated hyperedge representation.
• Figure 4: Data-driven Koopman operator dynamic learning module.
• Figure 5: Ablation Study.

Theorems & Definitions (1)

• Definition 1